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Motivated by the central role played by rotationally symmetric distributions in directional statistics, we consider the problem of testing rotational symmetry on the hypersphere. We adopt a semiparametric approach and tackle the situations where the location of the symmetry axis is either specified or unspecified. For each problem, we define two te...
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... consider magnetic remanence measurements made on samples collected from Paleozoic red-beds in Argentina. The data, that consists in n = 26 observations on S 2 , is showed in Figure 7. In line with the fact that the location θ θ θ is unknown a priori, Ley et al. (2013) considered the problem of estimating θ θ θ under the assumption of rotational symmetry. ...
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... may wonder, however, whether or not this assumption is appropriate in the present context. Visual inspection of Figure 7 indeed reveals that the density contours in the tangent space to the mode θ θ θ could be ellipses rather than circles. We therefore intend to test for rotational symmetry (about an unspecified θ θ θ) for the data at hand. ...
Citations
... Concepts of depth for directional data, i.e. unit vectors in R d , have been studied, for instance, by Liu et al. [12] and Pandolfo et al. [13]. Ley et al. [14] introduced quantiles for directional data and the angular Mahalanobis depth, and García-Portugués et al. [15] developed optimal tests for rotational symmetry against new classes of hyperspherical distributions. ...
... Future work could address the removal of disturbances or the highlighting of directional changes in DTI images by morphological operations [8,9]. (t + s) , π (15) = α t+s (i). ...
We define morphological operators and filters for directional images whose pixel values are unit vectors. This requires an ordering relation for unit vectors which is obtained by using depth functions. They provide a centre-outward ordering with respect to a specified centre vector. We apply our operators on synthetic directional images and compare them with classical morphological operators for grey-scale images. As application examples, we enhance the fault region in a compressed glass foam and segment misaligned fibre regions of glass fibre reinforced polymers.
... These tests all reject H 0 : P = P 0 (where P denotes the actual distribution of the observations, P 0 the distribution characterized by F 0 ) for large values of these distances. This is fine for real-valued observations Z of the Cramér-von Mises, Anderson-Darling, and Rothman type (García-Portugués et al. (2020; (ii) depth-and quantile-based tests based on angular simplicial depth and angular Tukey depth (Liu and Singh (1992), Rousseeuw and Struyf (2004) and Agostinelli and Romanazzi (2013)); tests based on the quantiles of sample projections onto the mean direction ( Ley et al. (2014), Mushkudiani (2002), Pandolfo et al. (2018); (iii) Kernel-based methods: (Hall et al. (1987) 2014)); (iv) rank-based methods (Ley et al. (2013(Ley et al. ( , 2017, Verdebout (2017)). ...
Measure transportation recently helped solve some long-standing open problems in statistical inference by providing satisfactory definitions of ranks and quantiles beyond the classical univariate case. We describe three examples of this fact: Wilcoxon-type distribution-free rank-based tests for the multivariate two-sample problem with unspecified error densities, nonparametric multiple-output quantile regression, and Cramér-von Mises Goodness-of-Fit tests for directional data.
... This kind of data is supported on the unit hypersphere S q := {x ∈ R q+1 : x = x x = 1}, q ≥ 1. Classical instances of "circular data" on the unit circle S 1 are given by wind direction and animal direction movements, yet less obvious appearances, such as the direction of cracks in medical hip prostheses (Mann et al., 2003;Ameijeiras-Alonso et al., 2021), are possible. The location of sunspots on the Sun's surface (García-Portugués et al., 2020) or the position of craters , among others, lie on S 2 and are examples of "spherical data" in astronomy. Applications involving directional data commonly appear in dimensions q = 1, 2. However, higher-dimensional directional data arises in text mining (Banerjee et al., 2005) or in genetics (Eisen et al., 1998). ...
... with Unif(S q ) denoting the uniform distribution on S q . Testing uniformity on S q is also the basis for further inference, such as testing spherical symmetry of distributions on R d , d ≥ 1 (e.g., Cai et al., 2013) or testing rotational symmetry about a direction µ ∈ S q (e.g., García-Portugués et al., 2020). Testing uniformity on S q is a problem that has been widely studied in the literature and still receives attention today. ...
Two new omnibus tests of uniformity for data on the hypersphere are proposed. The new test statistics leverage closed-form expressions for orthogonal polynomials, feature tuning parameters, and are related to a "smooth maximum" function and the Poisson kernel. We obtain exact moments of the test statistics under uniformity and rotationally symmetric alternatives, and give their null asymptotic distributions. We consider approximate oracle tuning parameters that maximize the power of the tests against generic alternatives and provide tests that estimate oracle parameters through cross-validated procedures while maintaining the significance level. Numerical experiments explore the effectiveness of null asymptotic distributions and the accuracy of inexpensive approximations of exact null distributions. A simulation study compares the powers of the new tests with other tests of the Sobolev class, showing the benefits of the former. The proposed tests are applied to the study of the (seemingly uniform) nursing times of wild polar bears.
... [55], [26]). Previous works on error-free circular, spherical or general hyperspherical data include density estimation ( [27], [23]), regression analysis ( [8], [51], [52], [32]) and statistical testing ( [11], [5], [24]). Among them, [8] did not cover a measurement error problem and simply considered the case where both response and predictor are spherical variables and the response is symmetrically distributed around the product of an unknown orthogonal matrix and the predictor. ...
... We analyzed the dataset 'sunspots births' in the R package 'rotasym' ( [25]). The dataset was analyzed in [24] to test the rotational symmetry of sunspots. Sunspots are temporary phenomena on the sun that appear as spots darker than the surrounding areas. ...
This paper studies density estimation and regression analysis with contaminated data observed on the unit hypersphere S^d. Our methodology and theory are based on harmonic analysis on general S^d. We establish novel nonparametric density and regression estimators, and study their asymptotic properties including the rates of convergence and asymptotic distributions. We also provide asymptotic confidence intervals based on the asymptotic distributions of the estimators and on the empirical likelihood technique. We present practical details on implementation as well as the results of numerical studies.
... Inference for the parameters of rotationally symmetric distributions has recently been considered in Christie (2015), Kanika et al. (2015), and Paindaveine and Verdebout (2020a,b). Extensions of rotational symmetry yielding, after projection in the tangent space (to S d −1 ) at θ θ θ, distributions with elliptic contours have been proposed in Kent (1982), Scealy and Wood (2019) and García-Portugués et al. (2020); see also Kume et al. (2013), Kume and Sei (2018) and Kent et al. (2018). ...
... (ii) the tangent vMF distribution as defined in García-Portugués et al. (2020). The tangent vMF distribution with location θ θ θ, angular function G , skewness direction µ, and skewness intensity κ is the distribution of ...
... The sunspots of the 22nd (September 1986 to July 1996; n 1 = 4551 points, in red) and 23rd (August 1996 to November 2008; n 2 = 5373 points, in green) solar cycles are shown in in Figure 5. Visual inspection hardly help decide whether these two samples are from the same distribution or not. According to García-Portugués et al. (2020), various tests suggest rotational symmetry around the north pole for the 23rd cycle while rejecting that hypothesis (p-values smaller than 0.02) for the 22nd cycle. ...
This paper proposes various nonparametric tools based on measure transportation for directional data. We use optimal transports to define new notions of distribution and quantile functions on the hypersphere, with meaningful quantile contours and regions yielding closed-form formulas under the classical assumption of rotational symmetry. The empirical versions of our distribution functions enjoy the expected Glivenko-Cantelli property of traditional distribution functions. They provide fully distribution-free concepts of ranks and signs and define data-driven systems of (curvilinear) parallels and (hyper)meridians. Based on this, we also construct a universally consistent test of uniformity which, in simulations, outperforms the ``projected'' Cram\' er-von Mises, Anderson-Darling, and Rothman procedures recently proposed in the literature. We also propose fully distribution-free rank- and sign-based tests for directional MANOVA. Two real-data examples involving the analysis of sunspots and proteins structures conclude the paper.
... Distributions on S d−1 which are rotationally symmetric about a direction µ ∈ S d−1 are also often regarded as the natural non-uniform arXiv:2210.06098v1 [math.ST] 12 Oct 2022 distributions on S d−1 [5]. In most cases, rotationally symmetric distributions have tractable normalising constants. ...
... where ω d−1 is the surface area of S d−2 [5]. A widely known distribution in R µ is the von Mises-Fisher distribution where f (t) = exp(κt). ...
We present canonical quantiles and depths for directional data following a distribution which is elliptically symmetric about a direction $\mu$ on the sphere $\mathcal{S}^{d-1}$. Our approach extends the concept of Ley et al. [1], which provides valuable geometric properties of the depth contours (such as convexity and rotational equivariance) and a Bahadur-type representation of the quantiles. Their concept is canonical for rotationally symmetric depth contours. However, it also produces rotationally symmetric depth contours when the underlying distribution is not rotationally symmetric. We solve this lack of flexibility for distributions with elliptical depth contours. The basic idea is to deform the elliptic contours by a diffeomorphic mapping to rotationally symmetric contours, thus reverting to the canonical case in Ley et al. [1]. A Monte Carlo simulation study confirms our results. We use our method to evaluate the ellipticity of depth contours and for trimming of directional data. The analysis of fibre directions in fibre-reinforced concrete underlines the practical relevance.
... (ii) tangent vMF distribution as in García-Portugués et al. (2020). A random vector X has a tangent vMF distribution with location µ, angular function G, skewness direction ω, skewness intensity κ iff X has the same distribution as ...
Non-Euclidean data is currently prevalent in many fields, necessitating the development of novel concepts such as distribution functions, quantiles, rankings, and signs for these data in order to conduct nonparametric statistical inference. This study provides new thoughts on quantiles, both locally and globally, in metric spaces. This is realized by expanding upon metric distribution function proposed by Wang et al. (2021). Rank and sign are defined at both the local and global levels as a natural consequence of the center-outward ordering of metric spaces brought about by the local and global quantiles. The theoretical properties are established, such as the root-$n$ consistency and uniform consistency of the local and global empirical quantiles and the distribution-freeness of ranks and signs. The empirical metric median, which is defined here as the 0th empirical global metric quantile, is proven to be resistant to contaminations by means of both theoretical and numerical approaches. Quantiles have been shown valuable through extensive simulations in a number of metric spaces. Moreover, we introduce a family of fast rank-based independence tests for a generic metric space. Monte Carlo experiments show good finite-sample performance of the test. Quantiles are demonstrated in a real-world setting by analysing hippocampal data.
... Let μ be a vector in S n , and let g be a function from [−1, 1] to [0, ∞). The pdf p(x|μ, g) dω n = c rs g(μ x) dω n , x ∈ S n is said to be rotationally symmetric [10]. For example, the von Mises-Fisher distribution is rotationally symmetric, with g(t) = exp(κt), κ ≥ 0. Garcia-Portugués et al. [10] define two tests for rotational symmetry, given the value of μ. ...
... The pdf p(x|μ, g) dω n = c rs g(μ x) dω n , x ∈ S n is said to be rotationally symmetric [10]. For example, the von Mises-Fisher distribution is rotationally symmetric, with g(t) = exp(κt), κ ≥ 0. Garcia-Portugués et al. [10] define two tests for rotational symmetry, given the value of μ. ...
... Garcia-Portugués et al. [10] define two extensions of the rotationally symmetric distributions, namely the tangent von Mises-Fisher distribution and the tangent elliptical distribution. Let μ be a vector in S n , and let μ be an (n + 1) × n matrix such that the columns are an orthonormal basis of the orthogonal complement of μ. ...
A family of probability density functions (pdfs) is defined on the unit hypersphere Sn. The parameter space for the pdfs is G(d,n+1)×R≥0, for 1≤d≤n, where G(d,n+1) is the Grassmannian of d-dimensional linear subspaces in Rn+1 and R≥0 is the range of values for a concentration parameter. This family of pdfs generalises the Watson distribution on the sphere S2. It is shown that the pdfs are tractable, in that (i) a given pdf can be sampled efficiently, (ii) the parameters of a pdf can be estimated using maximum likelihood, and (iii) the Kullback–Leibler divergence and the Fisher–Rao metric on G(d,n+1)×R≥0 have simple forms. A wide range of shapes of the pdfs can be obtained by varying d and the concentration parameter. The pdfs are used to model clusters of feature vectors on the hypersphere. The clusters are compared using the Kullback–Leibler divergences of the associated pdfs. Experiments with the mnist, Human Activity Recognition and Gas Sensor Array Drift datasets show that good results can be obtained from clustering algorithms based on the Kullback–Leibler divergence, even if the dimension n of the hypersphere is high.
... Concepts of depth for directional data, i.e. unit vectors in R d , have been studied, for instance, by Liu et al. [12] and Pandolfo et al. [13]. Ley et al. [14] introduced quantiles for directional data and the angular Mahalanobis depth, and García-Portugués et al. [15] developed optimal tests for rotational symmetry against new classes of hyperspherical distributions. ...
We define morphological operators and filters for directional images whose pixel values are vectors on the unit sphere. This requires an ordering relation for unit vectors which is obtained by using depth functions. They provide a centreoutward ordering with respect to a specified centre vector. We apply our operators on synthetic directional images and compare them with classical morphological operators for greyscale images. As application example, we enhance the fault region in a compressed glass foam.
... Sunspots observations have been well documented in the Debrecen Photoheliographic Data and the Greenwhich Photoheliographic Results. From the latter source, García-Portugués et al. (2020) give a curated dataset with the centers of groups of sunspots. The locations of the centers refer to the first-ever observations of such sunspots (henceforth referred to as "births"). ...
... S 1 is remarkably linear: a suitably-adapted linear fit to the grid of 100 points defining S 1 yields R 2 = 1 − 3 × 10 −5 . Therefore, the apparent linearity of the innovations in the series of sunspots births longitudes points towards the non-existence of (at least) major preferred longitudes during the 23rd solar cycle, a result that is coherent with the non-rejection of rotational symmetry for such cycle reported in García-Portugués et al. (2020). ...
A particularly challenging context for dimensionality reduction is multivariate circular data, i.e., data supported on a torus. Such kind of data appears, e.g., in the analysis of various phenomena in ecology and astronomy, as well as in molecular structures. This paper introduces Scaled Torus Principal Component Analysis (ST-PCA), a novel approach to perform dimensionality reduction with toroidal data. ST-PCA finds a data-driven map from a torus to a sphere of the same dimension and a certain radius. The map is constructed with multidimensional scaling to minimize the discrepancy between pairwise geodesic distances in both spaces. ST-PCA then resorts to principal nested spheres to obtain a nested sequence of subspheres that best fits the data, which can afterwards be inverted back to the torus. Numerical experiments illustrate how ST-PCA can be used to achieve meaningful dimensionality reduction on low-dimensional torii, particularly with the purpose of clusters separation, while two data applications in astronomy (three-dimensional torus) and molecular biology (on a seven-dimensional torus) show that ST-PCA outperforms existing methods for the investigated datasets.
